The Coefficients command extracts the coefficients of x in the polynomial p, where x and p can involve anticommutative variables. Coefficients is a one-command generalization of coeff, coeffs, lcoeff and tcoeff, that works with commutative noncommutative and anticommutative variables in equal footing.

The first argument, p, can also be a relation between polynomials, or a set or list of them, in which case Coefficients maps itself over the elements of the relation, set or list. For example, if p is an equation, then Coefficients(p, x) returns the equation obtained by computing Coefficients(lhs(p), x) = Coefficients(rhs(p), x), where lhs(p) and rhs(p) respectively represent the left and right hand sides of p.

The second argument, x, can be a name, function, product, power, or a list of them. When x is a power, say as in a^n, Coefficients(p, x) returns the same as Coefficients(p, a, n), that is the coefficient of the nth power.

The third argument, N, is optional, and indicates whether to extract all the coefficients (default behavior when N is not given, this is as coeffs) or the one of the Nth power when N is an integer (this is how coeff works), or the leading or trailing coefficient (pass N as the corresponding word, this produces the equivalent of lcoeff and tcoeff results), or a sequence of coefficients when N is a range of integers. The case N = all is then equivalent to the range case N = lower_degree .. higher_degree. When N is omitted it is assumed equal to all.

Unlike coeff and coeffs, when x is a single variable and more than one coefficient is requested (for example, you call Coefficients with just two arguments, or with a third argument as a range), Coefficients returns the sequence of coefficients in ascending order, including those that are equal to 0. To receive only the non-zero coefficients use the optional argument onlynonzero.

When x is a product, say a * b, Coefficients(p, a * b, N) returns the equivalent of taking a * b as a single variable - say c. Note: when a and b are anticommutative, Coefficients(p, a*b) returns the same as - Coefficients(p, b*a); likely, Coefficients(a*b, a) = - Coefficients(b*a, a).

When x is a list, say [a, b], Coefficients(p, [a, b], N) returns the equivalent of recursively computing the coefficients with respect to each of the elements of the list, that is the same as op(map(Coefficients, [Coefficients(p, a, N)], b, N)). Note that when a and b are anticommutative, Coefficients(p, [a, b]) returns the same as - Coefficients(p, [b, a]); likely, Coefficients(a*b, [a, b]) = - Coefficients(b*a, [a, b]).

Related to extracting coefficients, to compute the Degree of an expression with respect to anticommutative variables use the Physics:-Library:-Degree command.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\mathrm{\theta }\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{\theta }\right\}\right]$

$a\mathrm{\theta }\left[1\right]\mathrm{\theta }\left[2\right]+b$

$a{\mathrm{\theta }}_1{\mathrm{\theta }}_2+b$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\right)$

$b,a{\mathrm{\theta }}_2$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\,\mathrm{all}\right)$

$b,a{\mathrm{\theta }}_2$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\,0..1\right)$

$b,a{\mathrm{\theta }}_2$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\,0\right)$

$b$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\,1\right)$

$a{\mathrm{\theta }}_2$

$\mathrm{diff}\left(\,\mathrm{\theta }\left[1\right]\right)$

$a{\mathrm{\theta }}_2$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[2\right]\,1\right)$

$-a{\mathrm{\theta }}_1$

$\mathrm{diff}\left(\,\mathrm{\theta }\left[2\right]\right)$

$-a{\mathrm{\theta }}_1$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[1\right]\mathrm{\theta }\left[2\right]\right)$

$b,a$

$\mathrm{Coefficients}\left(\,\mathrm{\theta }\left[2\right]\mathrm{\theta }\left[1\right]\right)$

$b,-a$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\right)$

$b,0,0,a$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\,\mathrm{all}\right)$

$b,0,0,a$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\,\mathrm{onlynonzero}\right)$

$b,a$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\,\mathrm{leading}\right)$

$a$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[2\right]\,\mathrm{\theta }\left[1\right]\right]\,\mathrm{trailing}\right)$

$b$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\,0\right)$

$b$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\,1\right)$

$a$

$\mathrm{diff}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\right)$

$a$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[2\right]\,\mathrm{\theta }\left[1\right]\right]\,1\right)$

$-a$

$\mathrm{diff}\left(\,\left[\mathrm{\theta }\left[2\right]\,\mathrm{\theta }\left[1\right]\right]\right)$

$-a$

${\mathrm{\theta }\left[1\right]}^2$

$0$

$\mathrm{Coefficients}\left(\,\left[\mathrm{\theta }\left[1\right]\,\mathrm{\theta }\left[2\right]\right]\,2\right)$

$0$